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Finitely Presented Solvable Groups Conference

A 2-day conference at CCNY, focusing on the structure and properties of finitely presented solvable groups and on Gromov's view of groups as geometric objects.
When Mar 17, 2011 09:00 AM to
Mar 18, 2011 05:30 PM
Where City College of New York - Steinman Auditorium
Contact Name
Contact Phone 212.650.5167
Attendees Confirmed Speakers:
Gilbert Baumslag
Robert Bieri
Yves de Cornulier
John Groves
Olga Kharlampovich
Bruce Kleiner
Alexei Miasnikov
Walter D. Neumann
Irine Peng
Ralph Strebel
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In his address to the International Congress of Mathematicians in 1983, Gromov talked about groups as geometric objects. This followed on his proof in 1981 that finitely generated groups of polynomial growth contain a nilpotent subgroup of finite index, which has helped to focus attention on the extent to which the asymptotic properties of a finitely generated solvable group have on its algebraic structure.

This has led to a number of results dealing with the quasi-isometries and rigidity of solvable groups. Some of these results have been motivated by ideas coming out of the theory of Lie groups, where the semi-simple ones exhibit a rigidity that is not shared by the solvable ones. This ongoing geometric study of what are perhaps the simplest finitely generated solvable groups has put into sharp relief the very nature of finitely presented solvable groups. Although the work of Bieri and Strebel suggests that the structure of these finitely presented solvable groups is fairly restricted, very little is known in general about them, despite the existence of a number of negative algorithmic results. The special case of finitely presented metabelian groups, whose structure is closely connected to the structure of finitely generated modules over polynomial rings, algebraic geometry and seemingly also to Hilbert's Tenth Problem are themselves a particularly fascinating class of groups. There is an intriguing geometric invariant due to Bieri, Strebel and Neumann which distinguishes the finitely-presented metabelian groups from the finitely-generated, infinitely-related ones.

In view of the current situation, this suggests that a conference designed to join two rather different points of view, the geometric and the combinatorial, would lead to interesting new directions for research, joining these two points of view together. In addition, the possible connections with Hilbert's Tenth Problem provide the possibility of finding applications of recursive function theory to the study of finitely generated metabelian groups which are extremely promising.

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